Global Warming Basics: Trend Games

One of the things I emphasize most often and most strongly is that, while global temperature is forever fluctuating, it’s also showing a trend.

Temperature — be it local or global — is the sum of trend plus fluctuation. The trend is like the “rules of the game,” and it determines what we expect on average. That means it defines the climate, at least as far as temperature is concerned. The fluctuation is the “roll of the dice” — it sets the weather which, even when limited by the rules of the game, is still random and is not determined by the rules.

Politicians who want to avoid doing anything about global warming are very fond of saying that “climate is always changing.” Duh. In the past, climate has changed naturally, but in terms of global temperature those changes have generally been slow, happening on “geologic” time scales. For example, from about 18,000 years ago to around 10,000 years ago Earth warmed, dramatically on geologic time scales; the globe, on average, is about 5°C (9°F) hotter now than it was back then. For Earth as a whole, that’s a tremendous change — the difference between enjoying warm summer days at the beach in Chicago, and being covered by a miles-thick sheet of ice.

That warming, which opened the beaches of Lake Michigan to Chicagoans, took thousands of years. That’s natural climate change for you. By changing the conditions of the environment so, it made living conditions radically different. In order to survive, life had to adapt to the changing climate. It’s a good thing the change happened so slowly, because life adapts slowly.

Human life has enjoyed a remarkably stable climate for the last 10,000 years, a period called the “holocene.” Things have been stable enough for us to invent agriculture, domesticate dogs and horses (while cats domesticated us), build houses, build cities, domesticate livestock like cattle, pigs, and sheep, learn mettalurgy, invent writing, discover medicine and chemistry and physics, invent mathematics, master electronics, eventually invent the computer and the internet which binds us all into a global community.

Stable climate has been good to us; when the rules of the game remain the same, we learn to play by those rules and we do it very well.

Although global climate has been stable for mankind, local climate hasn’t always been so reliable. The small changes which have happened in certain regions have brought us trouble, sometimes disaster, sometimes bringing entire civilizations to their knees. Unstable climate has been very, very cruel to us.

When climate, and therefore temperature (which is part of weather and climate), is stable, the trend is flat; it doesn’t rise or fall. On a graph, it’s a flat line. Temperature at any moment, be it for the planet or some locale, can vary widely, but over the long haul it will fluctuate about some average which doesn’t change — that’s what makes the climate stable. If we observe Earth’s temperature during a stable climate, averaged over the entire globe, averaged over each of 100 years, it might look something like this:

This is just made-up data (using a random-number generator), consisting of a constant (stable climate) plus random noise (weather). It sure looks like temperature is stable in this example. But one of the great lessons of statistics is that looks can be deceiving. Therefore we can break out the big guns, the statistical tools, and do some heavy-duty mathematical analysis to look for a trend other than “not rising or falling.” In this case, the tests say “trend not there.” More precisely, they say “trend indistinguishable from zero.”

The estimated trend is actually -0.02 °C/century, cooling but very slowly — at that rate, it would take 5,000 years to cool by 1°C. But one of the things we get from the fancy math is the uncertainty of that estimate, which enables us to define what’s called a confidence interval. In this case the confidence interval is the range from -0.082 °C/century to +0.042 °C/century. Conclusion: it’s possible these made-up data show cooling, but not likely and not very fast; it’s also possible they show warming, but even less likely and again, not very fast. The mathematical way to say it is that because the confidence interval includes zero as a possibility, the estimated trend is not statistically significant.

Suppose instead that climate is warming, that there’s a rising trend in temperature at 1.7 °C/century. Then our observations might have looked like this:

Now it looks like there’s a rising trend, fer shure. If we break out the mathematical artillery, it says “Yeah. Fer Shure.”

What if we only looked at the last 15 years of the data, the part inside this red box?

There’s a lot less data to work with — only 15 years instead of 100. That makes statistical estimates far more uncertain. Using only that data, the confidence interval extends from -0.08 to +1.81 °C/century. If all we knew was the last 15 years of data, our conclusion would be: indistinguishable from zero. Because, not statistically significant.

The estimated trend is still upward, but only at 0.87 °C/century. But according to the confidence interval it might be below zero, so “not statistically significant.” More important, the idea that “there’s not really evidence of warming” is the right conclusion, if that’s the only data we have to go on.

Thing is, that’s not the only data we have to go on. There’s a lot more, which puts the recent data in context: what came before. It kind of makes the “not warming” idea look silly. As it should. We knew — with statistical significance — that it was warming; we should be asking whether there’s sufficient evidence to conclude that changed.

Using just the last 15 years of data, the confidence interval is from -0.08 °C/century to +1.81 °C/century. Yes that includes zero, but it also includes 1.7, the warming rate before that time, so we can’t conclude the warming rate is different than it was before. The proper conclusion is: “there’s not really evidence of a trend change.”

Yet many among those who want to deny the danger of man-made climate change will state the wrong conclusion, in an attempt to cast doubt on the reality of man-made climate change. They’ll begin their trend estimate at a time specifically chosen so that the confidence interval includes zero. You can always find such a start time, because as the number of data gets smaller and smaller, the confidence interval gets bigger and bigger, eventually wide enough to include zero. Then they’ll tout that time span as evidence that maybe we’re not warming the planet. To make it sound convincing, you have to ignore the context, i.e. what came before. That’s why they only graph a limited span of time: they don’t want you to see what happened before.

Quite recently the author of a post at the WUWT blog felt the need to tell us that there has been “No Statistically Significant Satellite Warming For 23 Years.” For those who don’t already know, WUWT is one of the most prominent blogs which denies the danger of man-made climate change. He shows you this:

This isn’t made-up data like in my illustrative example, it’s satellite data for the temperature in the lower troposphere (the bottom several miles of the atmosphere). The flat line has a slope which is the bottom of the confidence interval (basically, zero). His “point” is that maybe the slope is zero — not warming. Maybe it’s not warming — yay!

But he doesn’t even show the data that came before, the data that establish a trend which is upward, above zero, with statistical significance. He also doesn’t say (and maybe isn’t even aware) that the top of the confidence interval includes the trend that existed beforehand, so there’s really not sufficient evidence to conclude that the trend changed — but that’s a question you can only ask if you have access to data from beforehand. Which we do.

Maybe he actually believes it. Maybe he doesn’t realize that when you’re investigating a trend, whether or not it’s meaningful, you don’t get to ignore what you’ve already seen. If data since 1993 were all we had from satellite measurements, that’s all you can include. But we have more, and acting as though it doesn’t exist is bad practice.

Returning to my made-up sample data, ignoring context amounts to making a model of the trend that looks like this:

It completely isolates the two time spans — before and after — achieving not statistically significant for last 15 years. Take a good look at the trend line in red: it’s a discontinuous trend, what I often call a “broken” trend. When it comes to global temperature, a broken trend is non-physical. Perhaps the trend really did change, that’s physically sensible, but it should at least be continuous, without some sudden jump from one value to another. If we use math to fit a model which does include a trend change at that time, but requires it to be continuous, we get this:

It no longer even “looks like” the trend has recently been flat. And when we check the stats, the confidence interval for the last-15-years trend no longer includes zero; recent warming actaully is statistically significant.

Only by ignoring what came before, allowing for a non-physical “broken” trend, can you claim “not statistically significant” for the last 15 years. But those who deny the danger of man-made climate change do exactly that: they ignore context and create models with a broken trend. Maybe, perhaps even probably, they’re not aware of precisely what they’re doing and its nonsense nature. But given that they’re unaware of how to do it right — how much should we rely on their claims?

There’s another crucial aspect to their common habit of ignoring, even concealing, the data that put things in proper context. When they choose a time at which to start, they don’t do it because there’s a good laws-of-physic reason, or because they have solid statistical evidence that it’s when the trend really did change. They choose their start time specifically because it gives them the result they want. There’s a name for that process: it’s called cherry-picking. It has a profound effect on the statistics — you’re much, much more likely to get the result you want if you’re allowed to pick and choose from a large number of options (just like you’re much more likely to win the lottery if you buy a lot of lottery tickets, than if you only buy one).

Sometimes it’s done innocently by those who aren’t aware of what they’re doing or how it affects the reliability of their conclusions. Far too often, it’s done deliberately, by those who either know better, or should know better, just how much it poisons the validity of their analysis and how misleading it is. When they have an agenda, political or ideological, they’re willing. We shouldn’t be.

In case you’re wondering how this impacts real-world data rather than the made-up data I used for my examples, let’s see what happens when we take a close-up of the data used by the poster at WUWT, the satellite data for lower-troposphere temperature (TLT) from Remote Sensing Systems (RSS). Here’s all of it (as of this writing), not just the cherry-picked time span shown on WUWT:

The thick, straight red line shows the estimated trend when you use all the data. The thick broken blue line shows what you get if you split it into two pieces because you’re going to show the later stuff only and declare the trend is “not statistically significant.” Notice how, by breaking the trend at a pre-selected moment, we can make both segments seem to be warming more slowly than the entire time span. That’s some masterful cherry-picking.

If we allow a trend change at the moment the WUWT author selected, but require the trend to be continuous rather than broken, we get this:

I’ve plotted the “trend-change” line in blue as before, but this time I plotted it as a thick dashed line. That’s because it’s so close to the “no-trend-change” line (in red) that if I used a solid line, they’d be on top of each other and you wouldn’t be able to see them both.

This whole idea that Earth was warming but isn’t any more, that recent data show maybe there’s no more global warming, is a sham. Presenting evidence to support that requires the kind of trend games that make analysis unreliable, including non-physical broken trends, failure to show context, and especially careful selection of start times for the specific purpose of getting a desired result — cherry-picking, one of the most common and favorite techniques of those who deny the danger of mand-made climate change.

15 responses to “Global Warming Basics: Trend Games”

I agree with your observation that global air and ocean temperatures are increasing in a quasi-linear fashion over the last 50 years or more. I also agree with the premise that weather is short-term random noise expressed like rolling a set of dice while climate is the loading of the dice so that not all outcomes have equal probability. What disturbs me in many climate change debates is the fixation on “trends” which are always taken as linear regressions.

From a physical perspective there no reason that the observed environmental changes have to be linear EXCEPT for the fact that calculus tells us that each curve can locally be approximated by a linear slope, but the “local” here refers to the short time scale dominated by weather which brings us back to cherry-picking. The very fact that people can (and dishonestly do) pick their favorite time period to get the trends they want proves, that there is no linear trend, but a non-linear curve that over the last 150 years indicates increased warming at the longest time scales resolved by the records. Linear regressions do not predict the future, but carefully constructed dynamical models presenting dominant physics well enough do present the range of possible future change.

[Response: Certainly the trend in global temperature (and other climate variables) is not linear. But when we estimate it statistically, there are time spans over which it’s indistinguishable from linear (and the last 40 years or so is one of those time spans). So, when asked “How fast are we warming?” linear regression over a suitably chosen (not cherry-picked) time gives as good an answer as we’re able to get.

I also suggest that such purely statistical fits *do* have some predictive power (Taylor’s theorem), but I’ve emphasized more than once that it’s only short-term. If it’s rising now, and has been for forty years, it’s likely (not certain, but prediction never is) to keep rising over the next several years. But — as I’ve also emphasized — extrapolating even modestly into the future (decades or even less) is a fool’s errand; that should be done with physics, not statistics.

I guess we’re essentially in agreement, except about one thing: that “The very fact that people can (and dishonestly do) pick their favorite time period to get the trends they want proves, that there is no linear trend.” I was able to pick a time period to get the “trend I wanted” from artificial data which was *known* to have a linear trend. One shouldn’t underestimate the power of cherry-picking.]

We agree, even on the last point. Data originating from a linear process (such as your artificial data or data from a simple pendulum with small amplitude and small friction) does contain a true linear trend, I did not try to argue against that, but neither climate nor weather are linear processes except in the limits that Taylor’s theorem tells us. For climate this time scale is about 20 years, I recall from Ben Santer’s work a few years back.

I value your expositions on signal vs. noise and significance testing and steer my students towards the statistical thoroughness that you expose on these pages often and well. All serious students of weather, climate, and its policy implications should feel these statistical essentials in their bones. Sadly, few do.

[Response: How much harder it is for the average citizen, the voter! That’s the motive behind the “Global Warming Basics” posts: to raise the bar for John and Jane Doe.]

One thing that apparently confuses a lot of people is that short term phenomena like El Ninos are called “climate” variation when they should be called weather. This is one of the climate science denial memes that feeds into the phrase “climate is always changing”. If someone claims that moving between El Ninos and La Ninas is “climate change” then of course “climate” is always changing.

Just one of the ways that the dishonesty of climate science denial is perpetuated.

Another great post, but I do have a minor issue with the statement that life adapts slowly. I think it’s more accurate to say that life tends to adapt slowly to a gradual change and quickly to a rapid change. The population effects do of course become much more severe with higher rates of change (something we certainly need to worry about), and I guess individual species have something of a cap on their rate of adaptation linked to the length of their reproductive cycle.

I’m not really sure why that bothered me enough to respond to it; it’s way outside my area and pretty inconsequential to the rest of your post. Keep up the good work.

The vast majority of the heat in the Earth’s biosphere is in its oceans. The vast majority of AGW goes into the Earth’s water, including oceans and ice. The heat content in Earth’s water (including water vapor and ice) does help to drive the Earth’s weather. Sea level is a powerful measure of the heat in the Earth’s water systems. Sea level rise is a robust measure of AGW.

By looking only at atmospheric temperatures, one discards the all the data on the vast majority of the heat in Earth’s biosphere.. By looking at only satellite data, one discards the data on increasing latent heat in the atmosphere. Compared to the vast heat in the oceans, latent heat in the atmosphere may not be significant, but storms driven by latent heat are economically and socially significant. Atmospheric temperature rise does not include the heat absorbed as polar ice melts. Warming ice presents a surface temp at its local melting point to the atmosphere or the ocean even as it absorbs 80 cal/gram. Due to enthalpy of fusion, atmospheric temperature will not be a good measure of how much heat has been accumulated by AGW until all the polar ice has melted, and the oceans have come to full equilibrium

It is dumb to discard most of the data on heat flows in the biosphere when testing for AGW. The simple statement that “sea level is rising at __ mm per year” reflects robust measures of large heat flows. However, satellite data gets all the glory and the best expert analysis.

I interpret the Knappenberger-Michaels (Monckton?) graph with 0.0 trend on WUWT site as indicating an infinite increase of “global warming” on February 1999 and any time since but I’m not all fancy-schmancy with statistics and that.
18 years 1 month = 217 months 1996-10 – 2014-10 = 0.0 degrees / century (older, WUWT)
15 years 9 months = 189 months 1999-02 – 2014-10 = 1.2 degrees / century (more recent, my eye balls)
More recent is clearly warmer then, and 1.2 degrees / 0.0 degrees = infinity rate of increase on February 1999. I became an alarmatardalib when WUWT presented that evidence at me. I suppose you dry professor types are going to point out that a time series that can show a radically-varying selection of trends with small changes in start or end points is a time series that is incapable of showing any trend and only an idiot like me or Charlie Watts would try, but where’s the fun in that ?

barry I’m sure that Tamino will set you right, but what he likely meant is that *adding* the pre-1993 data to entire RSS dataset establishes the presence of a non-zero trend, not that 1979-1993 alone does that – which given the short timescale would likely be a cherry-pick anyway.

RSS data starts in 1979. No cherry-pick, that’s all the data there is for that set. One must refer to a different data set to find a pre-’93 period with statistically significant warming – but then you are comparing two different metrics – surface temps at 2 meters to lower trop temps at a few kilometers height.